2024 EJC P2 Q11

2024 EJC P2 Q11

13 marks

The heel stack of a running shoe refers to the thickness of the sole of the shoe at the point where the heel of the foot sits in the shoe.

A running shoe manufacturer makes a model called "SuperFly" and claims that the average heel stack is \(42.7\) millimetres. After reading some online reviews, the production manager wishes to test if the heel stack of the SuperFly has been overstated.

  1. Explain whether the manager should carry out a \(1\)-tail or \(2\)-tail test. State appropriate hypotheses for the test, defining any symbols you use.[3]

The production manager goes to the factory to measure a random sample of \(16\) shoes. The heel stacks, \(x\) millimetres, are summarised as follows.

\[n = 16 \qquad \sum (x - 42.7) = -6.4 \qquad \sum (x - 42.7)^2 = 13.2\]

  1. Explain in this context what is meant by a "random sample".[1]
  2. What assumption(s), if any, does the production manager need to make about the population distribution of the heel stacks?[1]

  3. Calculate unbiased estimates for the population mean and population variance of the heel stacks, and carry out the hypothesis test in part (a) at \(2.5\%\) level of significance.

    [5]

The manufacturer upgrades the machine used to produce the sole of the SuperFly. It is later established that the heel stack of the SuperFly has a variance of \(0.987\text{ millimetres}^2\). The production manager tests another random sample of \(20\) shoes to see whether the mean heel stack has changed, at \(2.5\%\) level of significance.

  1. With the same assumption(s) as in part (c), find the range of values of the sample mean, in millimetres to \(1\) decimal place, such that he can conclude that the mean heel stack has changed. (Answers obtained by trial and improvement from a calculator will obtain no marks.)

    [3]
Similar questions are unavailable for this question.
Answer:(a) 1-tail test. Because "overstated" suggests that the heel stack is less than what is claimed. Let \({\mu}\) be the population mean heel stack of the SuperFly. \({H_0 : \mu = 42.7}\), \({H_1 : \mu < 42.7}\) (b) Every shoe has an equal probability of being selected, and every shoe is selected independently of other shoes. (c) Need to assume that the population heel stack is normally distributed in order for sample mean to also be normally distributed, as sample size is too small to use Central Limit Theorem approximation. (d) \({\bar{x} = 42.3\text{;}\hspace{0.5em} s^2 = 0.709}\) (\(3\) s.f.) So we do not reject \({H_0}\) and conclude at \({2.5\%}\) level of significance that there is insufficient evidence that the heel stack is less than \({42.7\text{ millimetres}}\). (e) \({\bar{x} < 42.2}\) or \({\bar{x} > 43.2}\)

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